If is a strict 2-category (or at least strictly unital), then an icon is identical to an oplax natural transformation whose 1-cell components are identities. In general, there is a bijection between icons and such oplax natural transformations, obtained by pre- and post-composing with the unit constraints of . The name “icon” derives from this correspondence: it is an Identity Component Oplax Natural-transformation.
Icons have technical importance in the theory of 2-categories. For instance, there is no 2-category (or even 3-category) of 2-categories, functors, and lax or oplax transformations (even with modifications), but there is a 2-category of 2-categories, functors, and icons. (In fact, this 2-category is the 2-category of algebras for a certain 2-monad.)
Additionally, if monoidal categories are regarded as one-object 2-categories, then monoidal functors can be identified with 2-functors, and monoidal transformations can be identified with icons.